The Data

• Sexually transmitted diseases are infections that are spread during vaginal, oral, or anal intercourse. Although sometimes undetected, STDs can cause serious health problems in individuals and lead to reproductive issues. Here, we examine two types of bacterial infections that can be easily treated once diagnosed. In order to understand the factors that influence the chance of reinfection and hopefully decrease the cases in high-risk populations, the following data was analyzed. Time to reinfection is studied for three different groups: those infected with gonorrhea, those infected with chlamydia, and those infected with both. We also analyze various predictors to see if they significantly influence the survival probability.

• The predictors are as follows:

  • race (white/black)
  • marital status (divorced/separated, married, or single)
  • age
  • years of schooling
  • type of initial infection
  • number of partners
  • participating in oral sex in the last year and month
  • participating in rectal sex in the last year and month
  • presence of certain symptoms at time of initial infection
  • condom use
  • vaginal involvement at examination

• If we can understand the factors that lead to an increased risk of reinfection, we can utilize targeted preventive care and hopefully reduce the number of individuals who become infected.

data(std)
std

• We will clean the data to make this more interpretable.

Exploratory Data Analysis

• Our first step is to visualize a few of the variables in our data:

• Below we create two pie charts. The first shows the percentages for each type of initial infection. The second shows the percentages for the number of partners each patient had.

• Roughly 45% of the patients were initially infected by chlamydia, 16% of the patients were initially infected by gonorrhea; 39% of the patients were infected by both types of bacteria to start with. It looks like that infection by chlamydia was more commonly seen than infection by gonorrhea.

• About 70% of the patients reported to have 1 sex partner; 16.6% of them had 2 partners; 8% had no sex partner; it is very rare to have 3 or more sex partners; still, it is worth notice that one of the patients reported to have 19 sex partners.

• We are also interested in the frequency of symptoms for each type of initial infection among those patients who were reinfected. Overall, discharge is the most common symptom for all three types of initial infection.

• We can see that discharge is seen the most, regardless of infection type, while the least common symptom is symptoms involving the lymph nodes.

Checking Proportionality

• In order to build Cox model, we need to check that the data meets our proportional hazards assumptions. We will check this with Kaplan-Meier Curves and Hazard Ratios.

Kaplan-Meier Curves

• Before applying the Cox regression, we first test to ensure that the proportionality assumption was not violated for the base model that compares the three different types of infection. The Kaplan-Meier curves are plotted below, and it can be seen that at no point do the curves intersect.
• The survdiff() function results indicate that there is a difference in survival time for the three categories of initial infection.

## Call:
## survdiff(formula = surv_object ~ iinfct, data = std)
## 
##                    N Observed Expected (O-E)^2/E (O-E)^2/V
## iinfct=gonorrhea 140       73     54.5   6.28042   7.50617
## iinfct=chlamydia 396      135    153.0   2.12201   3.81136
## iinfct=both      341      139    139.5   0.00166   0.00278
## 
##  Chisq= 8.5  on 2 degrees of freedom, p= 0.01

Hazard Ratios

• The Hazard Ratio plot indicates proportional hazards after some initial spiking in the curves before time = 100 days.

Cumulative Hazards and CLogLog

• Further, the complementary log-log curves run parallel to each other after crossing once around time = 75 days.

• Once the initial Cox model was applied, the Schoenfeld residuals were plotted and there is no evidence of a non-random pattern that would suggest a violation of the assumptions.

## Call:
## coxph(formula = surv_object ~ iinfct, data = std)
## 
##   n= 877, number of events= 347 
## 
##                    coef exp(coef) se(coef)      z Pr(>|z|)   
## iinfctchlamydia -0.4202    0.6569   0.1457 -2.884  0.00393 **
## iinfctboth      -0.2980    0.7423   0.1450 -2.055  0.03984 * 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
##                 exp(coef) exp(-coef) lower .95 upper .95
## iinfctchlamydia    0.6569      1.522    0.4937    0.8741
## iinfctboth         0.7423      1.347    0.5587    0.9863
## 
## Concordance= 0.524  (se = 0.016 )
## Likelihood ratio test= 7.93  on 2 df,   p=0.02
## Wald test            = 8.37  on 2 df,   p=0.02
## Score (logrank) test = 8.46  on 2 df,   p=0.01

- The cox.zph function was then applied to the model which resulted in a p-value of 0.24 reinforcing that the proportional hazards assumption is maintained. Therefore, we chose to apply the Cox model which is elaborated on the next slide.

Model Selection

The Full model

• We start with a full model that includes all covariates and then simplify the model after testing each covariate for significance.

cox1 <-
  coxph(
    surv_object ~ iinfct + marital + race + os12m + os30d +
      rs12m + rs30d + abdpain + discharge + dysuria + condom + 
      itch + lesion + rash + lymph + vagina + dchexam + abnode +
      age + yschool + npartner,
    data = std
  )
summary(cox1)

## Call:
## coxph(formula = surv_object ~ iinfct + marital + race + os12m + 
##     os30d + rs12m + rs30d + abdpain + discharge + dysuria + condom + 
##     itch + lesion + rash + lymph + vagina + dchexam + abnode + 
##     age + yschool + npartner, data = std)
## 
##   n= 877, number of events= 347 
## 
##                      coef exp(coef)  se(coef)      z Pr(>|z|)   
## iinfctchlamydia -0.331853  0.717593  0.149264 -2.223  0.02620 * 
## iinfctboth      -0.263731  0.768180  0.149262 -1.767  0.07724 . 
## maritalMarried   0.054707  1.056231  0.431282  0.127  0.89906   
## maritalSingle    0.408462  1.504502  0.295237  1.384  0.16651   
## raceWhite       -0.112104  0.893951  0.141248 -0.794  0.42739   
## os12m1          -0.205985  0.813845  0.212025 -0.972  0.33129   
## os30d1          -0.337675  0.713427  0.238454 -1.416  0.15675   
## rs12m1           0.036907  1.037596  0.444944  0.083  0.93389   
## rs30d1          -0.194243  0.823458  0.565166 -0.344  0.73108   
## abdpain1         0.233865  1.263474  0.155288  1.506  0.13207   
## discharge1       0.113143  1.119792  0.114148  0.991  0.32159   
## dysuria1         0.162934  1.176960  0.155112  1.050  0.29352   
## condomnever     -0.263793  0.768133  0.119652 -2.205  0.02748 * 
## itch1           -0.149871  0.860819  0.154492 -0.970  0.33200   
## lesion1         -0.188159  0.828483  0.333106 -0.565  0.57217   
## rash1            0.003932  1.003940  0.392693  0.010  0.99201   
## lymph1          -0.030839  0.969632  0.547483 -0.056  0.95508   
## vagina1          0.349257  1.418013  0.174697  1.999  0.04559 * 
## dchexam1        -0.465508  0.627816  0.229914 -2.025  0.04290 * 
## abnode1          0.193753  1.213797  0.425224  0.456  0.64864   
## age              0.007855  1.007886  0.013891  0.565  0.57176   
## yschool         -0.127454  0.880334  0.039309 -3.242  0.00119 **
## npartner         0.076185  1.079162  0.054067  1.409  0.15881   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
##                 exp(coef) exp(-coef) lower .95 upper .95
## iinfctchlamydia    0.7176     1.3935    0.5356    0.9615
## iinfctboth         0.7682     1.3018    0.5733    1.0292
## maritalMarried     1.0562     0.9468    0.4536    2.4596
## maritalSingle      1.5045     0.6647    0.8435    2.6835
## raceWhite          0.8940     1.1186    0.6778    1.1791
## os12m1             0.8138     1.2287    0.5371    1.2332
## os30d1             0.7134     1.4017    0.4471    1.1385
## rs12m1             1.0376     0.9638    0.4338    2.4818
## rs30d1             0.8235     1.2144    0.2720    2.4929
## abdpain1           1.2635     0.7915    0.9319    1.7130
## discharge1         1.1198     0.8930    0.8953    1.4006
## dysuria1           1.1770     0.8496    0.8684    1.5951
## condomnever        0.7681     1.3019    0.6076    0.9711
## itch1              0.8608     1.1617    0.6359    1.1652
## lesion1            0.8285     1.2070    0.4313    1.5916
## rash1              1.0039     0.9961    0.4650    2.1675
## lymph1             0.9696     1.0313    0.3316    2.8355
## vagina1            1.4180     0.7052    1.0069    1.9970
## dchexam1           0.6278     1.5928    0.4001    0.9852
## abnode1            1.2138     0.8239    0.5275    2.7932
## age                1.0079     0.9922    0.9808    1.0357
## yschool            0.8803     1.1359    0.8151    0.9508
## npartner           1.0792     0.9266    0.9707    1.1998
## 
## Concordance= 0.634  (se = 0.016 )
## Likelihood ratio test= 73.29  on 23 df,   p=4e-07
## Wald test            = 69.84  on 23 df,   p=1e-06
## Score (logrank) test = 71.68  on 23 df,   p=7e-07

• We then use the drop1() function to examine how the dropping of each variable influences the AIC of the model.
• The summary of the cox model as well as the drop1() function indicated that initial infection type, condom use, vaginal involvement, discharge at exam, and years of school were significant.

Simplifying the Model

• From here, the goal is to remove extraneous variables and include models that are statistically significant and lower the AIC of the model.

• After creating a model with just these variables, condom use became less significant. However, no condom use is much more significant with a p-value of 0.14 versus a p-value of 0.80 for condom use sometimes. We create a model that drops the condom variable, but the AIC increases, suggesting that we should keep it. To see if recategorizing the values would lead to an increase in significance and decrease in AIC, we split condom use into two categories: never and use, where use includes individuals who sometimes use condoms and those who always use condoms. The new p-value for never using condoms was 0.009 and AIC decreased, so we kept the transformed variable in the model.

cox.model = coxph(surv_object ~ iinfct+condom+vagina
                  +dchexam+yschool, data = std)
summary(cox.model)

## Call:
## coxph(formula = surv_object ~ iinfct + condom + vagina + dchexam + 
##     yschool, data = std)
## 
##   n= 877, number of events= 347 
## 
##                     coef exp(coef) se(coef)      z Pr(>|z|)    
## iinfctchlamydia -0.38468   0.68067  0.14618 -2.632  0.00850 ** 
## iinfctboth      -0.24681   0.78129  0.14561 -1.695  0.09006 .  
## condomnever     -0.29779   0.74245  0.11551 -2.578  0.00994 ** 
## vagina1          0.40660   1.50171  0.16739  2.429  0.01514 *  
## dchexam1        -0.37042   0.69044  0.22123 -1.674  0.09405 .  
## yschool         -0.14357   0.86626  0.03332 -4.308 1.64e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
##                 exp(coef) exp(-coef) lower .95 upper .95
## iinfctchlamydia    0.6807     1.4692    0.5111    0.9065
## iinfctboth         0.7813     1.2799    0.5873    1.0393
## condomnever        0.7425     1.3469    0.5920    0.9311
## vagina1            1.5017     0.6659    1.0817    2.0848
## dchexam1           0.6904     1.4483    0.4475    1.0652
## yschool            0.8663     1.1544    0.8115    0.9247
## 
## Concordance= 0.603  (se = 0.017 )
## Likelihood ratio test= 45.32  on 6 df,   p=4e-08
## Wald test            = 46.51  on 6 df,   p=2e-08
## Score (logrank) test = 46.77  on 6 df,   p=2e-08

• With our reduced model, we still have an overall highly significant model supported by the Liklihood Ratio Test, Wald test, and the Score test.

• In the next slide, we will use residuals to address our considerations of stratification and transformation of quantitative variables.

Residuals

Martingale

Transformation of Quantitative to a Categorical Variable

• We will observe the Martingale residuals for the Years of Schooling variable to decide if a categorical transformation would be more appropriate.

• The martingale residual plot for years of school indicates a line with a consistent slightly negative slope, indicating that no transformation is needed.

Schoenfield

##          chisq df    p
## iinfct  3.5184  2 0.17
## condom  1.0539  1 0.30
## vagina  1.2835  1 0.26
## dchexam 0.4085  1 0.52
## yschool 0.0214  1 0.88
## GLOBAL  6.1176  6 0.41

• Each of the Schoenfeld residual plots fails to display a non-random pattern, meaning that each of the variables included in the model maintains the proportionality that we tested for prior to adding covariates to the model. Because of this, we see no reason for making a stratified Cox model for this dataset.

dfBeta

• From the dfBeta values, we will be able to identify outliers, which will be further discussed in the last slide.

Cox-Snell

• Lastly, we check how our model fits using the Cox-Snell residuals.

Since our standard 45 degree line fits the curve well, we can conclude that our final model is indeed a good fit.

Interpretting our Model

Our Best Model

• Using our final model with the five explanatory variables: initial infection type, condom use, vaginal involvement, discharge at exam, and years in school, we obtain the following coefficients:

## Call:
## coxph(formula = surv_object ~ iinfct + condom + vagina + dchexam + 
##     yschool, data = std)
## 
##   n= 877, number of events= 347 
## 
##                     coef exp(coef) se(coef)      z Pr(>|z|)    
## iinfctchlamydia -0.38468   0.68067  0.14618 -2.632  0.00850 ** 
## iinfctboth      -0.24681   0.78129  0.14561 -1.695  0.09006 .  
## condomnever     -0.29779   0.74245  0.11551 -2.578  0.00994 ** 
## vagina1          0.40660   1.50171  0.16739  2.429  0.01514 *  
## dchexam1        -0.37042   0.69044  0.22123 -1.674  0.09405 .  
## yschool         -0.14357   0.86626  0.03332 -4.308 1.64e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
##                 exp(coef) exp(-coef) lower .95 upper .95
## iinfctchlamydia    0.6807     1.4692    0.5111    0.9065
## iinfctboth         0.7813     1.2799    0.5873    1.0393
## condomnever        0.7425     1.3469    0.5920    0.9311
## vagina1            1.5017     0.6659    1.0817    2.0848
## dchexam1           0.6904     1.4483    0.4475    1.0652
## yschool            0.8663     1.1544    0.8115    0.9247
## 
## Concordance= 0.603  (se = 0.017 )
## Likelihood ratio test= 45.32  on 6 df,   p=4e-08
## Wald test            = 46.51  on 6 df,   p=2e-08
## Score (logrank) test = 46.77  on 6 df,   p=2e-08

• Therefore, patients with an initial infection of chlamydia are 32% less likely to become reinfected with an STD compared to those initially infected with gonorrhea. Similarly, patients who initially were infected with both STDs are 22% less likely to become reinfected. Counterintuitively, those in the study who stated they never used condoms were at lower risk of reinfection by about 25%. This observation will be further discussed in the next section. Vaginal involvement increases the probability of reinfection by about 50%. Also surprisingly, discharge at time of the exam decreases risk of reinfection by 30%. Lastly, for each year increase in education, the risk of reinfection decreases by 14%.

Outliers

• Although there were no obvious outliers for the condom use variable or initial infection type, the other variables had outliers that are summarized below:

  • Martingale Residuals: 11, 221, 690
  • Deviance Residuals: no unusual points
  • Initial infection: chlamydia Influence: 366, 211
  • Initial infection: both Influence: 366
  • Condom use or not Influence: 525
  • Vagina involvement or not Influence: 831, 525, 11
  • Discharge at exam Influence: 831, 859, 154
  • Year at school Influence: 294, 525, 635, 743

• Let’s observe the most frequent outliers, observations 366, 525, and 831.
• Each of these three individuals had a longer than expected survival time and was a censored observation.
  • Observation 11 was infected with both chlamydia and gonorrhea, which should lead to an increased risk of reinfection. However, this observation never became reinfected afer 1468 days in the study.
  • Observation number 366 was initially infected with gonorrhea, which based on our model has the highest risk for reinfection, used condoms, and had a relatively low level of education (yschool=9), but still had a relatively long time to reinfection.
  • Observation number 525 was initially infected with both chlamydia and gonorrhea, which should lead to an increased risk of reinfection. They also reported vaginal involvement during their exam and had a relatively low level of education (yschool=8), but was never reinfected and censored at 1005 days.
  • Observation number 831 had vaginal involvement and no discharge at exam, but was not reinfected and also censored.

Discussion

• Although we already went over the mathematical interpretation of our model, we want to also touch on some intuitive reasoning behind our coefficients. First, we saw earlier that each additional year of schooling leads to a further reduction in risk of reinfection. As individuals progress through the public schooling system, they are progressively taught more about sexual education and diseases. Increased awareness would intuitively lead to a decrease in risk of contraction because people become more aware of the dangers of STDs and how to prevent contracting them.

• The model showed that vaginal involvement considerably increased the risk of reinfection. Vaginal involvement leads to an increased susceptibility to infection since there is tissue damage. Hence, when patients experience vaginal involvement they should be made aware that they are at a higher risk of contracting an STD and hence should be especially careful, being sure to discuss STD and testing with their partners.

• Counterintuitively, our model suggests that never using condoms actually decreases risk of reinfection. There are a few hypotheses that could be tested with further studies to explain this relationship. First, it may be that those individuals who are not using condoms are only in exclusive relationships and hence have one consistent partner. Alternatively, it may be the case that those people not using condoms are more selective with their partners.

• The last variable to discuss is discharge. This variable may require further exploration because we are not given any additional information in terms of the type of discharge. Vaginal discharge is actually a fairly common occurrence and does not necessarily imply that there is an issue. It would be helpful to have more information about the type of discharge and if the patient regularly has vaginal discharge.

• The goal of this study was to analyze risk factors for reinfection of gonorrhea and chlamydia in order to inform the population and decrease the spread of STDs. From this model, we found that years of education is the most influential factor in determining risk of reinfection. Therefore, we suggest that sexual education be consistently addressed earlier in school in order to raise awareness and communicate the importance of safe sex practices and STD testing.

• Lastly, we noticed in the exploratory data analysis that our population is lacking racial diversity. In further studies, it would be interesting to gather data from a more diverse population, and also gather information about income level to see if there is any correlation between income and risk of infection.

Thank you!